Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra.

The aim of this article is to establish the existence of the solution of non-linear functional integral equations x (l , h) = U (l , h , x (l , h)) + F l , h , ∫ 0 l ∫ 0 h P (l , h , r , u , x (r , u)) d r d u , x (l , h) × G l , h , ∫ 0 a ∫ 0 a Q l , h , r , u , x (r , u) d r d u , x (l , h) of two variables, which is of the form of two operators in the setting of Banach algebra C [ 0 , a ] × [ 0 , a ] , a > 0. Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on C [ 0 , a ] × [ 0 , a ] and a fixed point theorem, which is a generalization of Darbo's fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example. [ABSTRACT FROM AUTHOR]